I am doing a calculation where I am estimating a value $\omega$ by a Taylor polynomial. I know that $\omega \cdot a = f(b)$ and thus I can estimate $\omega$ by $a \cdot T_n f(b) $ where $T_n$ is the Taylor series of nth order.
Now, two questions.
$\\ 1$ If I wanted to calculate the "maximum deviation" of my estimate, would I then want to calculate the "remainder" of my $T_n f(b)$... and, what, multiply it by $a$?
$2$ If the answer to above is yes, during the remainder calculation, I would need to calculate an upper bound on $f^{n+1} (c)$ for a given $c$ between $x$ and $b$ (where $x$ is the point the Taylor series is centered around, and $b$ was the point we needed to have it evaluated at in order to carry out an estimation). But... what if one wanted to calculate with really high orders, i.e. $T_{50} f(b)$. Would I then need to calculate the 51th derivative and find and upper bound on that?
Or is there some way around it? My $f(x)$ is arcsin and it's derivatives are increasing on [x,b].